Random, Stochastic, and Self-similar Equations

一系列广泛的数学方法将被用来增加对发生在自相似、分形和无序介质中的随机过程的长期和短期行为的理解。自相似狄利克雷形式、扩散和随机游动的存在性和唯一性将在广泛的一类分形上得到证明，包括作为群的极限空间出现的无限分歧分形。高斯和非高斯热核估计和绿色函数估计将研究无序系统，如自相似和随机分形。此外，概率工具将被开发，以研究非交换分析和广义微分几何的无序空间，进行局部狄利克雷形式。此外，该项目将有助于遍历理论的产品不一定独立的矩阵和它们的关系，局部性质的分形过程。本计画将提供具小随机扰动之微分及差分方程之李雅普诺夫指数之渐近公式，以及随机微分方程之李雅普诺夫指数之估计，并将其与随机微分方程之谱问题及分形及其他无序介质中之波传播有关，对研究无序介质中之过程有所贡献分形（fractals）在物理、化学、生物科学和工程学中有着广泛的应用。在渗流集群中的扩散过程，分形物体的振动，随机障碍物的通道中的信号传播，分形天线中的电磁波，海洋学中的Rossby波，金融市场模型，神经结构只是这些过程的许多例子中的几个。因此，该项目有助于数学，物理，生物科学和工程的整合。该项目将教育和研究与本科生结合起来。该项目更广泛的影响包括促进科学和工程领域的人力资源开发，扩大代表性不足群体的参与，以及加强研究和教育基础设施。

A wide array of mathematical methods will be used to increase the understanding of the long and short term behavior of random processes occurring in self-similar, fractal and disordered media. The existence and uniqueness of self-similar Dirichlet forms, diffusions, and random walks will be proved on a wide class of fractals, including infinitely ramified fractals appearing as limit spaces of groups. Gaussian and non-Gaussian heat kernel estimates and Green's function estimates will be studied on disordered systems, such as self-similar and random fractals. Furthermore, probabilistic tools will be developed to study non-commutative analysis on and generalized differential geometry of disordered spaces that carry a local Dirichlet form. In addition, the project will contribute to the ergodic theory of products of not necessarily independent matrices and their relation to local properties of processes on fractals. Asymptotic formulas for Lyapunov exponents of differential and difference equations with small random perturbations, and estimates of the Lyapunov exponents of stochastic differential equations will be obtained, and related to the spectral problems for stochastic differential equations and wave propagation in fractal and other disordered media.The project contributes to the study of processes in disordered media (fractals), which have many applications in physics, chemistry, biological sciences and engineering. Diffusion processes in percolation clusters, vibrations of fractal objects, signal propagation in channels with random obstacles, electro-magnetic waves in fractal antennae, Rossby waves in oceanography, models of financial markets, neural structures are just a few of many examples of such processes. Thus the project contributes to the integration of mathematics, physics, biological sciences and engineering. The project integrates education and research with undergraduate students. The broader impacts of the project include contributions to the development of human resources in science and engineering, expanding participation of underrepresented groups, and enhancing infrastructure for research and education.